In a large skillet, heat oil over medium-high. Add chicken - cook until the chicken is no longer pink, stirring to crumble, about 5 minutes. Add garlic and cook until fragrant, about 1 minute. Stir in sauce and pepper - reduce heat and simmer 5 minutes. Remove from the heat and stir in fresh basil. Set aside to cool for 10 minutes.

Meanwhile, divide dough into 4 even pieces - stretch and roll each piece into an 8" x 6" rectangle on a lightly floured surface. Divide chicken mixture evenly between the rectangles - sprinkle each with a portion of the cheese. With each assembled rectangle, fold dough in half over filling, pinching edges to seal. Use a sharp knife to cut two or three slits on top of each assembled calzone.

Slide calzones onto the baking stone and bake until golden, about 10 to 15 minutes. Remove and let cool slightly before serving.